Lecture 4

Probability Distribution

Continuous Case

Definition: A random variable that can take on any value in an

interval is called continuous.

Definition: Let Y be any r.v. The distribution function of Y, F(y),

is such that .

Example: Y ~ Bin(2, ½). Find F(y).

Solution:

Properties of Distribution Function:

1.

2.

3. F(y) is a non-decreasing function of y

Definition: A r.v. Y with distribution function F(y) is said to be

continuous if F(y) is continuous, for .

Note: If Y is a continuous r.v. then for any .

Definition: Let F(y) be the distribution function for a continuous

r.v. Y. Then

(wherever the derivative exists),

is called the probability density function for Y.

Properties of Density Function:

1.

2. ∫

Example: Given {

, find f(y).

Solution:

Example: Let Y be a continuous r.v. with

{

.

Find F(y).

Solution:

Theorem: If Y is a continuous r.v. with f(y) and a < b, then

∫

.

Proof:

Example: (#4.16) {

is the density

function for Y.

(a) Find c and F(y);

(b) Graph f(y) and F(y);

(c) Find .

Solution:

Expected Value

Definition: The expected value of a continuous r.v. Y is

∫

(provided ∫ | |

)

Theorem: Let g(Y) be a function of Y, then

[ ] ∫

(provided ∫ | |

)

Properties of Expected Value:

1. E(c) = c, c is a constant.

2. E[cg(Y)]=cE[g(Y)], g(Y) is a function of Y.

3. [ ] [ ] [ ],

are functions of Y.

Proof:

Example: Given {

, find E(Y) and Var(Y).

Solution:

Uniform Distribution

Definition: If , a r.v. Y is said to have a continuous

uniform probability distribution on the interval if and

only if the density function of Y is {

.

Theorem: If Y ~ Unif , then

and

.

Proof:

Example: (#4.44) The change in depth of a river from one day to

the next, measured at a specific location is a r.v. Y with

{

(a) Find k;

(b) What’s the distribution function of Y?

Solution:

Example: (#4.50) Beginning at 12:00 am, a computer center is up

for 1 hour and then down for 2 hours on a regular cycle. A person

who is unaware of the schedule dials the center at a random time

between 12:00 am and 5:00 am. Find P(center is up when call

comes in).

Solution:

Normal Distribution

Definition: A r.v. Y is said to have a normal probability distribution

if and only if, for and ,

√

.

Theorem: If , then and .

Proof: later.

Example: Z~N(0, 1)

(a) Find ;

(b) Find .

Solution:

Example: (#4.68) The grade point averages (GPAs) of a large

population of students are appr. Normally distributed with

and . What fraction of the students will possess a GPA

greater than 30?

Solution:

Example: (#4.73) The width of bolts of fabric is normally

distributed with and . What is the

probability that a randomly chosen bolt has a width of between 947

and 958?

Solution:

Gamma Distribution

Definition: A r.v. Y is said to have a gamma distribution with

parameters and if and only if

{

Where ∫

is the gamma function.

Properties of Gamma Function:

Theorem: If , then and

.

Proof:

Definition: Let . A r.v. Y is said to have a Chi-square ( )

distribution with degrees of freedom if and only if

.

Theorem: If , then and .

Proof:

Definition: A r.v. Y is said to have an exponential distribution

with parameter if and only if .

Theorem: If , then and .

Proof:

Example: (#4.88) The magnitude of earthquakes recorded in a

region of North America can be modeled as having an exponential

distribution with mean 2.4. Find the probability that an earthquake

striking this region will (a) exceed 3.0; (b) fall between 2.0 and

3.0.

Solution:

Example: (#4.96) Given {

(a) Find k;

(b) Does Y have a -distribution? If so, what is ?

(c) Find E(Y) and Var(Y).

Solution:

Moments and Moment-Generating Functions

Definition: If Y is a continuous r.v., then the moment about

the origin is , k=1,2,…

The moment about the mean, or the central moment, is

, k=1,2,…

Example: Find for .

Solution:

Definition: If Y is a continuous r.v., then the moment-generating

function (mgf) of Y is

.

The mgf exists if there is a constant b > 0 such that m(t) is finite

for | | .

Example: Let . Find mgf for Y.

Solution:

Theorem: Let Y be a r.v. with density function f(y) and g(Y) be a

function of Y. Then the mgf of g(Y) is

[ ] ∫

.

Example: Let , . Find mgf for g(Y).

Solution:

Example: (#4.137) Show that if Y is a r.v. with mgf m(t) and U is

given by U = aY + b, then the mgf of U is .. Find E(U)

and Var(U), given that and .

Solution:

Example: (#4.138) Let . Find .

Solution: